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/** This class contains equations about the properties of water.
It is primarily based on The International Association for the Properties
of Water and Steam (IAPWS, http://iapws.org/) document IAPWS R6-95(2018),
a PDF of which is usually available from:
http://www.iapws.org/relguide/IAPWS-95.html
Specifically, the file
http://www.iapws.org/relguide/IAPWS95-2018.pdf
All references to equation and table numbers are the equation numbers in
this document.
The development of this document is discussed in
https://aip.scitation.org/doi/10.1063/1.1461829
There is an online calculator that can be used to validate these equations
at:
http://twt.mpei.ac.ru/mcs/worksheets/iapws/IAPWS95.xmcd
IAPWS also produces a simpler formulation "for industry", called IAPWS-IF97
which is described at:
http://www.iapws.org/relguide/IF97-Rev.html
That page contains a PDF of IAPWS-IF97 and some "backward equations" that
make it easier to calculate without iterating inverse solutions. These
may be used to find good "first guesses" for inverse solutions.
Simpler equations for vapor pressure and densities at the saturation point
are found here:
http://www.iapws.org/relguide/Supp-sat.html
especially in the PDF file:
http://www.iapws.org/relguide/Supp-sat.pdf
These can be used to obtain good initial guesses for finding
higher-accuracy results in the full IAPWS95 model.
A scientific paper describing the derivation of the saturation properties
can be found at:
https://aip.scitation.org/doi/10.1063/1.555926
All of the methods in this class are class-level methods so you don't
need an instance of the Water class to call them. Just call them like:
Water.boilingPoint[1 atmosphere]
*/
class Water
{
/** Constants as defined in IAPWS RS-95(2018). These may not exactly match
the current best-known values of these constants in the SI, but they are
used as defined there for purposes of matching its output exactly.
*/
/** The temperature of the critical point of water. (eq.1) */
class var Tc = 647.096 K
/** The critical density of water. (eq.2) */
class var rhoc = 322 kg m^-3
/** The specific gas constant. Due to the use of the specific gas constant,
Eq. (4) corresponds to a mass-based formulation. In order to convert
values of specific properties to molar properties, an appropriate value
for the molar mass should be used (eq. 3) */
class var R = 0.46151805 kJ kg^-1 K^-1
/** The critical pressure of water from IAPWS SR1-86(1992) */
class var pc = 22.064 MPa
/** The triple point of water. */
class var TriplePoint = 273.16 K
/** IAPWS doesn't directly define the molar mass of water directly but
indicates that the best-known values of several of these constants have
changed since the model was published. It sort of waves its hand at
George S. Kell, Effects of isotopic composition, temperature, pressure,
and dissolved gases on the density of liquid water Journal of Physical
and Chemical Reference Data 6, 1109 (1977);
https://doi.org/10.1063/1.555561
to hint that it contains a suitable definition of the molar mass. Its
masses and isotope ratios are taken from "Atomic Weights of the
Elements 1973", Pure Appl. Chem 37, 589-603 (1974) Of course, none of
this matches the current definition of Avogadro's constant, the gas
constant, etc. Anyway, this is the version of the molar mass of water
defined in that paper, and the one that should probably be used with
this model and its definition of R. See the notes for R above. */
class var molarMassH2O = 18.015242 g/mol
/** Calculates the pressure at the specified density and temperature.
p = rho^2 (df/drho)_T
See Table 3 for this equation.
*/
class pressure[rho is mass_density, T is temperature] :=
{
delta = rho / rhoc
return rho R T (1 + delta dphirdeltaDimensioned[rho, T])
}
/** Calculates the compressibility factor Z = p / (rho R T) */
class compressibilityFactor[rho is mass_density, T is temperature] :=
{
return pressure[rho, T] / (rho R T)
}
/** Calculates the specific internal energy of the system at the specified
density and temperature. Multiply this by mass to get the energy.
u = f - T (df/dT)_rho
returns energy / mass
*/
class specificInternalEnergy[rho is mass_density, T is temperature] :=
{
delta = rho / rhoc
tau = Tc / T
return tau R T dphidtau[delta, tau]
}
/** Calculates the specific entropy
s = -(df/dT)_rho
Multiply by mass to get the total entropy.
This has units of specific heat capacity (e.g. cal/g/degC)
*/
class specificEntropy[rho is mass_density, T is temperature] :=
{
delta = rho / rhoc
tau = Tc / T
return R (tau dphidtau[delta, tau] - phi[delta, tau])
}
/** Calculates the specific enthalpy
h = f - T(df/dT)_rho + rho(df/drho)_T
Multiply by mass to get the total enthalpy.
This has units of energy/mass
*/
class specificEnthalpy[rho is mass_density, T is temperature] :=
{
delta = rho / rhoc
tau = Tc / T
return R T (1 + tau dphidtau[delta, tau] + delta dphirdelta[delta, tau])
}
/** Calculates the specific isochoric heat capacity
c_v = (du/dT)_rho
This has units of specific heat capacity (e.g. cal/g/degC)
*/
class specificIsochoricHeatCapacity[rho is mass_density, T is temperature] :=
{
delta = rho / rhoc
tau = Tc / T
return R (-tau^2 d2phitau[delta, tau])
}
/** Calculates the specific isobaric heat capacity
c_p = (dh/dT)_p
This has units of specific heat capacity (e.g. cal/g/degC)
*/
class specificIsobaricHeatCapacity[rho is mass_density, T is temperature] :=
{
delta = rho / rhoc
tau = Tc / T
return R (-tau^2 d2phitau[delta, tau] + (1 + delta dphirdelta[delta, tau] - delta tau d2phirdeltatau[delta, tau])^2 / (1 + 2 delta dphirdelta[delta, tau] + delta^2 d2phirdelta[delta, tau]))
}
/** Calculates the speed of sound.
w = (dp/drho)^(1/2)
This has units of velocity
*/
class speedOfSound[rho is mass_density, T is temperature] :=
{
delta = rho / rhoc
tau = Tc / T
return sqrt[R T (1 + 2 delta dphirdelta[delta, tau] + delta^2 d2phirdelta[delta,tau] - (1 + delta dphirdelta[delta, tau] - delta tau d2phirdeltatau[delta, tau])^2/(tau^2 (d2phitau[delta, tau])))]
}
/** The Joule-Thompson coefficient
mu = (dT/drho)_h
Results have units of temperature / pressure
*/
class JouleThompsonCoefficient[rho is mass_density, T is temperature] :=
{
delta = rho / rhoc
tau = Tc / T
return (-(delta dphirdelta[delta, tau] + delta^2 d2phirdelta[delta,tau] + delta tau d2phirdeltatau[delta,tau]) / ((1 + delta dphirdelta[delta, tau] - delta tau d2phirdeltatau[delta,tau])^2 - tau^2 d2phitau[delta,tau] (1 + 2 delta dphirdelta[delta, tau] + delta^2 d2phirdelta[delta, tau]))) / (R rho)
}
/** The isentropic temperature-pressure coefficent
beta_s = (dT/dp)_s
Results have units of temperature / pressure
*/
class isentropicTemperaturePressureCoefficient[rho is mass_density, T is temperature] :=
{
delta = rho / rhoc
tau = Tc / T
return (1 + delta dphirdelta[delta, tau] - delta tau d2phirdeltatau[delta, tau]) / ((1 + delta dphirdelta[delta, tau] - delta tau d2phirdeltatau[delta,tau])^2 - tau^2 d2phitau[delta,tau] (1 + 2 delta dphirdelta[delta, tau] + delta^2 d2phirdelta[delta, tau])) / (R rho)
}
/** The isothermal throttling coefficient
delta_T = (dh/dp)_T
This has units of inverse density (e.g. m^3 / kg)
*/
class isothermalThrottlingCoefficient[rho is mass_density, T is temperature] :=
{
delta = rho / rhoc
tau = Tc / T
return (1-(1 + delta dphirdelta[delta, tau] - delta tau d2phirdeltatau[delta, tau])/ (1 + 2 delta dphirdelta[delta, tau] + delta^2 d2phirdelta[delta,tau])) / rho
}
/** The second virial coefficient B(T).
"Even at low density, real gases don't quite obey the ideal gas law.
A systematic way to account for deviations from ideal behavior is the
virial expansion,
P V = n R T (1 + B(T)/(V/n) + C(T)/(V/n)^2 + ...)
where the functions B(T), C(T), and so on are called the "virial
coefficents". When the density of the gas is fairly low, so that the
volume per mole is large, each term in the series is much smaller than
the one before. In many situations it's sufficient to omit the third
term and concentrate on the second, whose coefficent B(T) is called the
second virial coefficient (the first coefficient being 1.)"
--Daniel V. Schroeder, An Introduction to Thermal Physics, p.9
B(tau) rhoc = lim[phirdelta(delta, tau), delta->0]
Units are m^3 kg^-1
This actually returns the *specific* version of B.
To get the dimensionless units that will work in the equation above,
you need to multiply by Water.molarMassH2O
*/
class B[T is temperature] :=
{
tau = Tc / T
sum = 0
for i = [1,2,3,8,9,10,23]
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni tau^ti
}
for i = 55 to 56
{
[ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i
thetai = Ai + 1 - tau // See footnote b, p.12
sum = sum + ni (thetai^2 + Bi)^bi exp[-Ci - Di(tau-1)^2]
}
return sum/rhoc
}
/** The third virial coefficient C(T). See the discussion for the second
virial coefficient B[T] above.
Units are m^6 kg^-2
This actually returns the *specific* version of C.
To get the dimensionless units for the equation documented in B[T],
you need to multiply by Water.molarMassH2O
*/
class C[T is temperature] :=
{
tau = Tc / T
sum = 0
for i = [4,5,11,12,24,25,26]
{
[ci, di, ti, ni] = Table2@i
sum = sum + 2 ni tau^ti
}
for i = 8 to 10
{
[ci, di, ti, ni] = Table2@i
sum = sum - 2 ni tau^ti
}
for i = 55 to 56
{
[ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i
thetai = Ai + 1 - tau // See footnote b, p.12
sum = sum + 4 ni (Ci (thetai^2 + Bi) - bi (Ai thetai / betai + Bi ai))* (thetai^2 + Bi)^(bi-1) exp[-Ci - Di(tau-1)^2]
}
return sum/rhoc^2
}
/** Calculates the specific internal energy of the system at the specified
atmospheric pressure equals the saturated vapor pressure. You can use
StandardAtmosphereTest.frink which calls StandardAtmosphere.frink
to find the approximate atmospheric pressure for any altitude.
*/
class boilingPoint[ps is pressure] :=
{
return saturatedTemperature[ps]
}
/** Returns the approximate saturated vapor pressure of water from
IAPWS SR1-86(1992). This does not use the full IAPWS95 model, but it
can be used to get a very good first guess. This is Section 3 */
class saturatedVaporPressure[T is temperature] :=
{
theta = T/Tc
tau = 1 - theta
p = pc exp[Tc/T (-7.85951783 tau +
1.84408259 tau^(3/2) +
-11.7866497 tau^3 +
22.6807411 tau^(7/2) +
-15.9618719 tau^4 +
1.80122502 tau^(15/2))]
return p
}
/** Returns the approximate saturated vapor pressure of water from
IAPWS IF-97. This does not use the full IAPWS95 model, but it
can be used to get a very good first guess. This is Section 8 and
8.1 of IAPWS IF-97 */
class saturatedVaporPressure97[Ts is temperature] :=
{
Tstar = 1 K
Theta = Ts / Tstar + Table34@9 / ((Ts/Tstar) - Table34@10) // eq. 29b
A = Theta^2 + Table34@1 Theta + Table34@2
B = Table34@3 Theta^2 + Table34@4 Theta + Table34@5
C = Table34@6 Theta^2 + Table34@7 Theta + Table34@8
p = 1 MPa ((2 C) / (-B + (B^2 - 4 A C)^(1/2)))^4 // eq. 30
return p
}
/** Returns the approximate saturated vapor temperature of water from
IAPWS97. This does not use the full IAPWS95 model, but it
can be used to get a very good first guess. This is Section 8.2 of
IAPWS97. */
class saturatedTemperature[ps is pressure] :=
{
// Equation 31
pstar = 1 MPa
beta = (ps / pstar)^(1/4) // Eq. 29a
E = beta^2 + Table34@3 beta + Table34@6
F = Table34@1 beta^2 + Table34@4 beta + Table34@7
G = Table34@2 beta^2 + Table34@5 beta + Table34@8
D = (2 G)/(-F - (F^2 - 4 E G)^(1/2))
Ts = 1 K ((Table34@10 + D - ((Table34@10 + D)^2 - 4(Table34@9 + Table34@10 D))^(1/2)) / 2)
return Ts
}
/** Returns the approximate saturated liquid density of water from
IAPWS SR1-86(1992). This does not use the full IAPWS95 model, but it
can be used to get a very good first guess. This is Section 4.1 */
class saturatedLiquidDensity[T is temperature] :=
{
theta = T/Tc
tau = 1-theta
rho = rhoc (1 + 1.99274064 tau^(1/3) + // b1
1.09965342 tau^(2/3) + // b2
-0.510839303 tau^(5/3) + // b3
-1.75493479 tau^(16/3) + // b4
-45.5170352 tau^(43/3) + // b5
-6.74694450e5 tau^(110/3)) // b6
return rho
}
/** Returns the approximate saturated vapor density of water from
IAPWS SR1-86(1992). This does not use the full IAPWS95 model, but it
can be used to get a very good first guess. This is Section 4.2 */
class saturatedVaporDensity[T is temperature] :=
{
theta = T/Tc
tau = 1-theta
rho = rhoc exp[1 + -2.03150240 tau^(2/6) + // c1
-2.68302940 tau^(4/6) + // c2
-5.38626492 tau^(8/6) + // c3
-17.2991605 tau^(18/6) + // c4
-44.7586581 tau^(37/6) + // c5
-63.9201063 tau^(71/6)] // c6
return rho
}
/** The specific Helmholtz free energy f (also sometimes called A).
The IAPWS formulation is implemented as a fundamental equation for the
specific Helmholtz free energy f.
This equation is expressed in dimensionless form, phi = f/(R T),
or, conversely f = phi R T
(see eq. 4 and following)
*/
class f[rho is mass_density, T is temperature, debug=false] :=
{
return R T phiDimensioned[rho, T, debug]
}
/** The formulation of the phi equation is implemented as a dimensionless
version phi[delta, tau] where
delta = rho / rhoc
tau = Tc / T
This turns a dimensionally-correct call with density and temperature
into the internal dimensionless form. (see eq.4 and following)
*/
class phiDimensioned[rho is mass_density, T is temperature, debug=false] :=
{
return phi[rho / rhoc, Tc / T, debug]
}
/** The dimensionless phi function phi[delta, tau] is separated into two
parts, an ideal-gas part phi0[delta, tau] and a residual part
phir[delta, tau] (eq. 4)
*/
class phi[delta is dimensionless, tau is dimensionless, debug=false] :=
{
phi0 = phi0[delta, tau]
phir = phir[delta, tau]
if debug
println["phi0=$phi0\nphir=$phir\n"]
return phi0 + phir
}
/** Calculates the first derivative of phi with respect to tau. This
includes both the ideal gas and residual parts. */
class dphidtau[delta is dimensionless, tau is dimensionless] :=
{
return dphi0dtau[delta, tau] + dphirtau[delta, tau]
}
/** Calculates the second derivative of phi with respect to tau. This
includes both the ideal gas and residual parts. */
class d2phitau[delta is dimensionless, tau is dimensionless] :=
{
return d2phi0dtau[delta, tau] + d2phirtau[delta, tau]
}
/** Calculates the second derivative of phi with respect to delta. This
includes both the ideal gas and residual parts. */
class d2phidelta[delta is dimensionless, tau is dimensionless] :=
{
return d2phi0ddelta[delta, tau] + d2phirdelta[delta, tau]
}
/** The ideal gas part phi0 of the dimensionless Helmholtz free energy.
This is obtained from an equation for the specific isobaric heat
capacity in the ideal-gas state developed by Cooper [6]
This implements eq. 5
*/
class phi0[delta is dimensionless, tau is dimensionless] :=
{
sum = ln[delta] + Table1@1 + /* n1 0 */
Table1@2 tau + /* n2 0 */
Table1@3 ln[tau] /* n3 0 */
for i = 4 to 8
{
[ni, gammai] = Table1@i
sum = sum + ni ln[1 - exp[-gammai tau]]
}
return sum
}
/** The first partial derivative of the ideal gas part of the dimensionless
Helmholtz free energy with respect to delta. */
class dphi0ddelta[delta is dimensionless, tau is dimensionless] :=
{
return 1/delta
}
/** The second partial derivative of the ideal gas part of the dimensionless
Helmholtz free energy with respect to delta. */
class d2phi0ddelta[delta is dimensionless, tau is dimensionless] :=
{
return -1 / delta^2
}
/** The first partial derivative of the ideal gas part of the dimensionless
Helmholtz free energy with respect to tau. */
class dphi0dtau[delta is dimensionless, tau is dimensionless] :=
{
sum = Table1@2 + Table1@3 / tau
for i=4 to 8
{
[ni, gammai] = Table1@i
sum = sum + ni gammai ((1 - exp[-gammai tau])^-1 - 1)
}
return sum
}
/** The second partial derivative of the ideal gas part of the dimensionless
Helmholtz free energy with respect to tau. */
class d2phi0dtau[delta is dimensionless, tau is dimensionless] :=
{
sum = -Table1@3 / tau^2
for i=4 to 8
{
[ni, gammai] = Table1@i
sum = sum - ni gammai^2 exp[-gammai tau] (1 - exp[-gammai tau])^-2
}
return sum
}
/** The second partial derivative of the ideal gas part of the dimensionless
Helmholtz free energy with respect to delta then with respect to tau. */
class d2phi0ddeltadtau[delta is dimensionless, tau is dimensionless] :=
{
return 0
}
/** The residual part phir[delta, tau] of the dimensionless Helmholtz free
energy.
This implements eq. 6
*/
class phir[delta is dimensionless, tau is dimensionless] :=
{
sum = 0
for i=1 to 7
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni delta^di tau^ti
}
for i=8 to 51
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni delta^di tau^ti e^(-delta^ci)
}
for i=52 to 54
{
[ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i
sum = sum + ni delta^di tau^ti * exp[-alphai (delta - epsiloni)^2 - betai(tau - gammai)^2]
}
for i=55 to 56
{
[ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i
Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2]
theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai))
Delta = theta^2 + Bi ((delta-1)^2)^ai
sum = sum + ni Delta^bi delta Psi
}
return sum
}
/** The partial derivative of phir[delta, tau] with respect to delta, with
correct dimensions. */
class dphirdeltaDimensioned[rho is mass_density, T is temperature] :=
{
return dphirdelta[rho / rhoc, Tc / T]
}
/** The partial derivative of phir[delta, tau] with respect to delta.
This implements eq. 6
*/
class dphirdelta[delta is dimensionless, tau is dimensionless] :=
{
sum = 0
for i=1 to 7
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni di delta^(di-1) tau^ti
}
for i=8 to 51
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni exp[-delta^ci] (delta^(di-1) tau^ti(di - ci delta^ci))
}
for i=52 to 54
{
[ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i
sum = sum + ni delta^di tau^ti exp[-alphai (delta-epsiloni)^2 - betai (tau-gammai)^2] (di/delta - 2 alphai (delta - epsiloni))
}
for i=55 to 56
{
[ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i
Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2]
theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai))
Delta = theta^2 + Bi ((delta-1)^2)^ai
dPsiddelta = -2 Ci (delta-1) Psi
dDeltaddelta = (delta-1) (Ai theta 2 / betai ((delta-1)^2)^(1/(2 betai)-1) + 2 Bi ai((delta-1)^2)^(ai-1))
dDeltabiddelta = bi Delta^(bi-1) dDeltaddelta
sum = sum + ni (Delta^bi (Psi + delta dPsiddelta) + dDeltabiddelta delta Psi)
}
return sum
}
/** The second partial derivative of phir[delta, tau] with respect to delta,
with correct dimensions. */
class d2phirdeltaDimensioned[rho is mass_density, T is temperature] :=
{
return d2phirdelta[rho / rhoc, Tc / T]
}
/** The second partial derivative of phir[delta, tau] with respect to delta.
*/
class d2phirdelta[delta is dimensionless, tau is dimensionless] :=
{
sum = 0
for i=1 to 7
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni di (di-1) delta^(di-2) tau^ti
}
for i=8 to 51
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni exp[-delta^ci] (delta^(di-2) tau^ti ((di - ci delta^ci)(di-1-ci delta^ci) - ci^2 delta^ci))
}
for i=52 to 54
{
[ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i
sum = sum + ni tau^ti exp[-alphai (delta-epsiloni)^2 - betai(tau-gammai)^2] * (-2 alphai delta^di + 4 alphai^2 delta^di (delta-epsiloni)^2 - 4 di alphai delta^(di-1) (delta-epsiloni) + di (di-1) delta^(di-2))
}
for i=55 to 56
{
[ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i
Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2]
theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai))
Delta = theta^2 + Bi ((delta-1)^2)^ai
dPsiddelta = -2 Ci (delta-1) Psi
dDeltaddelta = (delta-1) (Ai theta 2 / betai ((delta-1)^2)^(1/(2 betai)-1) + 2 Bi ai((delta-1)^2)^(ai-1))
d2Psiddelta = (2 Ci (delta-1)^2 -1) 2 Ci Psi
dDeltabiddelta = bi Delta^(bi-1) dDeltaddelta
d2Deltaddelta = 1 / (delta-1) dDeltaddelta + (delta-1)^2 (4 Bi ai (ai-1)((delta-1)^2)^(ai-2) + 2 Ai^2 (1/betai)^2 (((delta-1)^2)^(1/(2 betai) - 1))^2 + Ai theta 4/betai (1 / (2 betai) - 1)((delta-1)^2)^(1/(2 betai) - 2))
d2Deltabiddelta = bi (Delta^(bi-1) d2Deltaddelta + (bi - 1) Delta^(bi-2) dDeltaddelta^2)
sum = sum + ni (delta^bi (2 dPsiddelta + delta d2Psiddelta) + 2 dDeltabiddelta (Psi + delta dPsiddelta) + d2Deltabiddelta delta Psi)
}
return sum
}
/** The partial derivative of phir[delta, tau] with respect to tau, with
correct dimensions. */
class dphirtauDimensioned[rho is mass_density, T is temperature] :=
{
return dphirtau[rho / rhoc, Tc / T]
}
/** The partial derivative of phir[delta, tau] with respect to tau.
*/
class dphirtau[delta is dimensionless, tau is dimensionless] :=
{
sum = 0
for i=1 to 7
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni ti delta^di tau^(ti-1)
}
for i=8 to 51
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni ti delta^di tau^(ti-1) exp[-delta^ci]
}
for i=52 to 54
{
[ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i
sum = sum + ni delta^di tau^ti exp[-alphai (delta-epsiloni)^2 - betai (tau-gammai)^2] (ti/tau - 2 betai(tau - gammai))
}
for i=55 to 56
{
[ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i
Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2]
theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai))
Delta = theta^2 + Bi ((delta-1)^2)^ai
dDeltabidtau = -2 theta bi Delta^(bi-1)
dPsidtau = -2 Di (tau-1) Psi
sum = sum + ni Delta (dDeltabidtau Psi + Delta^bi dPsidtau)
}
return sum
}
/** The partial second derivative of phir[delta, tau] with respect to tau,
with correct dimensions. */
class d2phirtauDimensioned[rho is mass_density, T is temperature] :=
{
return d2phirtau[rho / rhoc, Tc / T]
}
/** The partial second derivative of phir[delta, tau] with respect to tau.
*/
class d2phirtau[delta is dimensionless, tau is dimensionless] :=
{
sum = 0
for i=1 to 7
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni ti (ti-1) delta^di tau^(ti-2)
}
for i=8 to 51
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni ti (ti-1) delta^di tau^(ti-2) exp[-delta^ci]
}
for i=52 to 54
{
[ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i
sum = sum + ni delta^di tau^ti exp[-alphai (delta-epsiloni)^2 - betai (tau-gammai)^2] ((ti/tau - 2 betai(tau - gammai))^2 - ti/tau^2 - 2 betai)
}
for i=55 to 56
{
[ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i
Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2]
theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai))
Delta = theta^2 + Bi ((delta-1)^2)^ai
dDeltabidtau = -2 theta bi Delta^(bi-1)
dPsidtau = -2 Di (tau-1) Psi
d2Psidtau = (2 Di (tau-1)^2 - 1) 2 Di Psi
d2Deltabidtau = 2 bi Delta^(bi-1) + 4 theta^2 bi (bi-1) Delta^(bi-2)
sum = sum + ni delta (d2Deltabidtau Psi + 2 dDeltabidtau dPsidtau + Delta^bi d2Psidtau)
}
return sum
}
/** The partial second derivative of phir[delta, tau] with respect to tau,
with correct dimensions. */
class d2phirdeltatauDimensioned[rho is mass_density, T is temperature] :=
{
return d2phirdeltatau[rho / rhoc, Tc / T]
}
/** The partial second derivative of phir[delta, tau] with respect to
delta then to tau.
*/
class d2phirdeltatau[delta is dimensionless, tau is dimensionless] :=
{
sum = 0
for i=1 to 7
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni di ti delta^(di-1) tau^(ti-1)
}
for i=8 to 51
{
[ci, di, ti, ni] = Table2@i
sum = sum + ni ti delta^(di-1) tau^(ti-1) (di - ci delta^ci) exp[-delta^ci]
}
for i=52 to 54
{
[ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i
sum = sum + ni delta^di tau^ti exp[-alphai (delta-epsiloni)^2 - betai (tau-gammai)^2] (di/delta - 2 alphai (delta - epsiloni))(ti/tau - 2 betai(tau - gammai))
}
for i=55 to 56
{
[ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i
Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2]
theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai))
Delta = theta^2 + Bi ((delta-1)^2)^ai
dDeltabidtau = -2 theta bi Delta^(bi-1)
dPsidtau = -2 Di (tau-1) Psi
d2Psidtau = (2 Di (tau-1)^2 - 1) 2 Di Psi
dPsiddelta = -2 Ci (delta-1) Psi
d2Deltabidtau = 2 bi Delta^(bi-1) + 4 theta^2 bi (bi-1) Delta^(bi-2)
dDeltaddelta = (delta-1) (Ai theta 2 / betai ((delta-1)^2)^(1/(2 betai)-1) + 2 Bi ai((delta-1)^2)^(ai-1))
dDeltabiddelta = bi Delta^(bi-1) dDeltaddelta
d2Psiddeltadtau = 4 Ci Di (delta-1)(tau-1) Psi
d2Deltabiddeltadtau = -Ai bi 2 / betai Delta^(bi-1) (delta-1)((delta-1)^2)^(1/(2 betai) - 1) - 2 theta bi (bi-1) Delta^(bi-2) dDeltaddelta
sum = sum + ni (Delta^(bi) (dPsidtau + delta d2Psiddeltadtau) + delta dDeltabiddelta dPsidtau + dDeltabidtau (Psi + delta dPsiddelta) + d2Deltabiddeltadtau delta Psi)
}
return sum
}
/** Numerical values of the coefficients and parameters of the ideal-gas
part of the dimensionless Helmholtz free energy, Eq. 5. This is
table 1 in IAPWS-95.
The coluns are [ni0, gammai0]
*/
class var Table1 = [undef, // No element 0
-8.3204464837497,
6.6832105275932,
3.00632,
[0.012436, 1.28728967],
[0.97315, 3.53734222],
[1.27950, 7.74073708],
[0.96956, 9.24437796],
[0.24873, 27.5075105]]
/** Numerical values of the coefficients and parameters of the
residual part of the dimensionless Helmholtz free energy, Eq. (6) */
class var Table2 = [undef, // No element 0
//[ci, di, ti, ni]
[x, 1, -1/2, 0.12533547935523e-1],
[x, 1, 7/8, 0.78957634722828e1],
[x, 1, 1, -0.87803203303561e1],
[x, 2, 1/2, 0.31802509345418],
[x, 2, 3/4, -0.26145533859358],
[x, 3, 3/8, -0.78199751687981e-2],
[x, 4, 1, 0.88089493102134e-2],
[1, 1, 4, -0.66856572307965],
[1, 1, 6, 0.20433810950965],
[1, 1, 12, -0.66212605039687e-4],
[1, 2, 1, -0.19232721156002],
[1, 2, 5, -0.25709043003438],
[1, 3, 4, 0.16074868486251],
[1, 4, 2, -0.40092828925807e-1],
[1, 4, 13, 0.39343422603254e-6],
[1, 5, 9, -0.75941377088144e-5],
[1, 7, 3, 0.56250979351888e-3],
[1, 9, 4, -0.15608652257135e-4],
[1, 10, 11, 0.11537996422951e-8],
[1, 11, 4, 0.36582165144204e-6],
[1, 13, 13, -0.13251180074668e-11],
[1, 15, 1, -0.62639586912454e-9],
[2, 1, 7, -0.10793600908932],
[2, 2, 1, 0.17611491008752e-1],
[2, 2, 9, 0.22132295167546],
[2, 2, 10, -0.40247669763528],
[2, 3, 10, 0.58083399985759],
[2, 4, 3, 0.49969146990806e-2],
[2, 4, 7, -0.31358700712549e-1],
[2, 4, 10, -0.74315929710341],
[2, 5, 10, 0.47807329915480],
[2, 6, 6, 0.20527940895948e-1],
[2, 6, 10, -0.13636435110343],
[2, 7, 10, 0.14180634400617e-1],
[2, 9, 1, 0.83326504880713e-2],
[2, 9, 2, -0.29052336009585e-1],
[2, 9, 3, 0.38615085574206e-1],
[2, 9, 4, -0.20393486513704e-1],
[2, 9, 8, -0.16554050063734e-2],
[2, 10, 6, 0.19955571979541e-2],
[2, 10, 9, 0.15870308324157e-3],
[2, 12, 8, -0.16388568342530e-4],
[3, 3, 16, 0.43613615723811e-1],
[3, 4, 22, 0.34994005463765e-1],
[3, 4, 23, -0.76788197844621e-1],
[3, 5, 23, 0.22446277332006e-1],
[4, 14, 10, -0.62689710414685e-4],
[6, 3, 50, -0.55711118565645e-9],
[6, 6, 44, -0.19905718354408],
[6, 6, 46, 0.31777497330738],
[6, 6, 50, -0.11841182425981],
// [ci, di, ti, ni, alphai, betai, gammai, epsiloni]
[x, 3, 0, -0.31306260323435e2, 20, 150, 1.21, 1],
[x, 3, 1, 0.31546140237781e2, 20, 150, 1.21, 1],
[x, 3, 4, -0.25213154341695e4, 20, 250, 1.25, 1],
// [ai, bi, Bi, ni, Ci, Di, Ai, betai]
[7/2, 85/100, 0.2, -0.14874640856724, 28, 700, 0.32, 3/10],
[7/2, 95/100, 0.2, 0.31806110878444, 32, 800, 0.32, 3/10]]
/** Numerical values for IAPWS97 table 34. These are used to solve
saturation-pressure and saturation-temperature equations. */
class var Table34 = [undef, // No n0
0.11670521452767e4, // n1
-0.72421316703206e6, // n2
-0.17073846940092e2, // n3
0.12020824702470e5, // n4
-0.32325550322333e7, // n5
0.14915108613530e2, // n6
-0.48232657361591e4, // n7
0.40511340542057e6, // n8
-0.23855557567849, // n9
0.65017534844798e3] // n10
}
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Alan Eliasen was born 20217 days, 23 hours, 52 minutes ago.